Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have in mind a theorem. I think it's straightforward but tedious to prove carefully and I'd like to use it to answer another question. Has anyone heard of this theorem? Does anyone know of a quick (non-tedious!) proof?

Suppose $(\Omega,\Sigma,\mathbb{P})$ is a probability space. Suppose $\mathcal{P}_1,\mathcal{P}_2,\ldots$ is a sequence of increasingly fine finite partitions of $\Omega$. That is, each $\mathcal{P}_n$ is a finite partition of $\Omega$ and if $A\in\mathcal{P}_{n+1}$, then $A\subset A'$ for some $A'\in\mathcal{P}_n$. And the big assumption: Suppose for each $\epsilon>0$, there is an $n$ such that $\mathbb{P}(A)<\epsilon$ for all $A\in\mathcal{P}_n$. Then for any possible cumulative distribution function $F:\mathbb{R}\to[0,1]$ (i.e., increasing, right continuous, $F(-\infty)=0$, $F(\infty)=1$), there is a random variable $X$ on $(\Omega,\Sigma,\mathbb{P})$ with $F$ as its cdf.

I am wondering if there's a middle ground between "it can be shown" and a proof by direct construction, which is tedious.

share|cite|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.